Estatística

Teste de Aderência

Prof. Antonio Fernando Branco Costa

e-mail: fbranco@feg.unesp.br

Página da FEG: www.feg.unesp.br/~fbranco

Teste de Aderência

IDÉIA: descobrir qual é a Distribuição de uma Variável Aleatória X, a partir de uma amostra: {X1, X2, ..., Xn}

A partir da amostra, existe evidência estatística para afirmar que o dado é honesto, ou seja, que X tem Distribuição Equiprovável ???

Problema:

Seja X: nº que sai na jogada de um dado

i 1 2 3 4 5 6 Totalfi = Oi 185 239 206 188 174 208 1200

TESTE DE HIPÓTESES: H0: X tem Distribuição EquiprovávelH1: Tal não ocorre

i 1 2 3 4 5 6 Totalfi = Oi 200 200 200 200 200 200 1200

Ei 200 200 200 200 200 200 1200Oi - Ei 0 0 0 0 0 0 0

Caso Perfeito

Logo, espera-se que em 1200 jogadas saia 200 vezes cada número:

X equiprovável então 61 i)Pr(Xpi

200611200pnE ii

i 1 2 3 4 5 6 Totalfi = Oi 185 239 206 188 174 208 1200

Ei 200 200 200 200 200 200 1200Oi - Ei -15 39 6 -12 -26 8

O que ocorre

TESTE DE HIPÓTESES: H0: X tem Distribuição EquiprovávelH1: Tal não ocorre

i 1 2 3 4 5 6 Totalfi = Oi 185 239 206 188 174 208 1200

Ei 200 200 200 200 200 200 1200Oi - Ei -15 39 6 -12 -26 8 0

1,13 7,61 0,18 0,72 3,38 0,32 13,33

i

2ii

E)E(O

2

1Calculado

k

i

i

2ii

E)E(O

Critério: Rejeitar H0 se grande!for2Calculado

m1k ν

k : número de classes

m : número de parâmetros estimados,

5ν 016

Distribuições - valores de , onde P = P( )2 2,Pv

2,

2Pvv

0,995 0,99 0,975 0,95 0,90 0,75 0,50 0,25 0,10 0,05 0,025 0,01 0,005 0,0011 3,93E-05 1,57E-04 9,82E-04 3,93E-03 0,0158 0,102 0,455 1,323 2,706 3,841 5,024 6,635 7,879 10,8282 0,0100 0,0201 0,0506 0,103 0,211 0,575 1,386 2,773 4,605 5,991 7,378 9,210 10,597 13,8163 0,0717 0,115 0,216 0,352 0,584 1,213 2,366 4,108 6,251 7,815 9,348 11,345 12,838 16,2664 0,207 0,297 0,484 0,711 1,064 1,923 3,357 5,385 7,779 9,488 11,143 13,277 14,860 18,4675 0,412 0,554 0,831 1,145 1,610 2,675 4,351 6,626 9,236 11,070 12,833 15,086 16,750 20,5156 0,676 0,872 1,237 1,635 2,204 3,455 5,348 7,841 10,645 12,592 14,449 16,812 18,548 22,4587 0,989 1,239 1,690 2,167 2,833 4,255 6,346 9,037 12,017 14,067 16,013 18,475 20,278 24,3228 1,344 1,646 2,180 2,733 3,490 5,071 7,344 10,219 13,362 15,507 17,535 20,090 21,955 26,1249 1,735 2,088 2,700 3,325 4,168 5,899 8,343 11,389 14,684 16,919 19,023 21,666 23,589 27,877

10 2,156 2,558 3,247 3,940 4,865 6,737 9,342 12,549 15,987 18,307 20,483 23,209 25,188 29,58811 2,603 3,053 3,816 4,575 5,578 7,584 10,341 13,701 17,275 19,675 21,920 24,725 26,757 31,26412 3,074 3,571 4,404 5,226 6,304 8,438 11,340 14,845 18,549 21,026 23,337 26,217 28,300 32,90913 3,565 4,107 5,009 5,892 7,042 9,299 12,340 15,984 19,812 22,362 24,736 27,688 29,819 34,52814 4,075 4,660 5,629 6,571 7,790 10,165 13,339 17,117 21,064 23,685 26,119 29,141 31,319 36,12315 4,601 5,229 6,262 7,261 8,547 11,037 14,339 18,245 22,307 24,996 27,488 30,578 32,801 37,69716 5,142 5,812 6,908 7,962 9,312 11,912 15,338 19,369 23,542 26,296 28,845 32,000 34,267 39,25217 5,697 6,408 7,564 8,672 10,085 12,792 16,338 20,489 24,769 27,587 30,191 33,409 35,718 40,79018 6,265 7,015 8,231 9,390 10,865 13,675 17,338 21,605 25,989 28,869 31,526 34,805 37,156 42,31219 6,844 7,633 8,907 10,117 11,651 14,562 18,338 22,718 27,204 30,144 32,852 36,191 38,582 43,82020 7,434 8,260 9,591 10,851 12,443 15,452 19,337 23,828 28,412 31,410 34,170 37,566 39,997 45,31521 8,034 8,897 10,283 11,591 13,240 16,344 20,337 24,935 29,615 32,671 35,479 38,932 41,401 46,79722 8,643 9,542 10,982 12,338 14,041 17,240 21,337 26,039 30,813 33,924 36,781 40,289 42,796 48,26823 9,260 10,196 11,689 13,091 14,848 18,137 22,337 27,141 32,007 35,172 38,076 41,638 44,181 49,72824 9,886 10,856 12,401 13,848 15,659 19,037 23,337 28,241 33,196 36,415 39,364 42,980 45,559 51,17925 10,520 11,524 13,120 14,611 16,473 19,939 24,337 29,339 34,382 37,652 40,646 44,314 46,928 52,62026 11,160 12,198 13,844 15,379 17,292 20,843 25,336 30,435 35,563 38,885 41,923 45,642 48,290 54,05227 11,808 12,879 14,573 16,151 18,114 21,749 26,336 31,528 36,741 40,113 43,195 46,963 49,645 55,47628 12,461 13,565 15,308 16,928 18,939 22,657 27,336 32,620 37,916 41,337 44,461 48,278 50,993 56,89229 13,121 14,256 16,047 17,708 19,768 23,567 28,336 33,711 39,087 42,557 45,722 49,588 52,336 58,30130 13,787 14,953 16,791 18,493 20,599 24,478 29,336 34,800 40,256 43,773 46,979 50,892 53,672 59,70340 20,707 22,164 24,433 26,509 29,051 33,660 39,335 45,616 51,805 55,758 59,342 63,691 66,766 73,40250 27,991 29,707 32,357 34,764 37,689 42,942 49,335 56,334 63,167 67,505 71,420 76,154 79,490 86,66160 35,534 37,485 40,482 43,188 46,459 52,294 59,335 66,981 74,397 79,082 83,298 88,379 91,952 99,607

Pv 0,995 0,99 0,975 0,95 0,90 0,75 0,50 0,25 0,10 0,05 0,025 0,01 0,005 0,0011 3,93E-05 1,57E-04 9,82E-04 3,93E-03 0,0158 0,102 0,455 1,323 2,706 3,841 5,024 6,635 7,879 10,8282 0,0100 0,0201 0,0506 0,103 0,211 0,575 1,386 2,773 4,605 5,991 7,378 9,210 10,597 13,8163 0,0717 0,115 0,216 0,352 0,584 1,213 2,366 4,108 6,251 7,815 9,348 11,345 12,838 16,2664 0,207 0,297 0,484 0,711 1,064 1,923 3,357 5,385 7,779 9,488 11,143 13,277 14,860 18,4675 0,412 0,554 0,831 1,145 1,610 2,675 4,351 6,626 9,236 11,070 12,833 15,086 16,750 20,5156 0,676 0,872 1,237 1,635 2,204 3,455 5,348 7,841 10,645 12,592 14,449 16,812 18,548 22,4587 0,989 1,239 1,690 2,167 2,833 4,255 6,346 9,037 12,017 14,067 16,013 18,475 20,278 24,3228 1,344 1,646 2,180 2,733 3,490 5,071 7,344 10,219 13,362 15,507 17,535 20,090 21,955 26,1249 1,735 2,088 2,700 3,325 4,168 5,899 8,343 11,389 14,684 16,919 19,023 21,666 23,589 27,87710 2,156 2,558 3,247 3,940 4,865 6,737 9,342 12,549 15,987 18,307 20,483 23,209 25,188 29,58811 2,603 3,053 3,816 4,575 5,578 7,584 10,341 13,701 17,275 19,675 21,920 24,725 26,757 31,26412 3,074 3,571 4,404 5,226 6,304 8,438 11,340 14,845 18,549 21,026 23,337 26,217 28,300 32,90913 3,565 4,107 5,009 5,892 7,042 9,299 12,340 15,984 19,812 22,362 24,736 27,688 29,819 34,52814 4,075 4,660 5,629 6,571 7,790 10,165 13,339 17,117 21,064 23,685 26,119 29,141 31,319 36,12315 4,601 5,229 6,262 7,261 8,547 11,037 14,339 18,245 22,307 24,996 27,488 30,578 32,801 37,69716 5,142 5,812 6,908 7,962 9,312 11,912 15,338 19,369 23,542 26,296 28,845 32,000 34,267 39,25217 5,697 6,408 7,564 8,672 10,085 12,792 16,338 20,489 24,769 27,587 30,191 33,409 35,718 40,79018 6,265 7,015 8,231 9,390 10,865 13,675 17,338 21,605 25,989 28,869 31,526 34,805 37,156 42,31219 6,844 7,633 8,907 10,117 11,651 14,562 18,338 22,718 27,204 30,144 32,852 36,191 38,582 43,82020 7,434 8,260 9,591 10,851 12,443 15,452 19,337 23,828 28,412 31,410 34,170 37,566 39,997 45,31521 8,034 8,897 10,283 11,591 13,240 16,344 20,337 24,935 29,615 32,671 35,479 38,932 41,401 46,79722 8,643 9,542 10,982 12,338 14,041 17,240 21,337 26,039 30,813 33,924 36,781 40,289 42,796 48,26823 9,260 10,196 11,689 13,091 14,848 18,137 22,337 27,141 32,007 35,172 38,076 41,638 44,181 49,72824 9,886 10,856 12,401 13,848 15,659 19,037 23,337 28,241 33,196 36,415 39,364 42,980 45,559 51,17925 10,520 11,524 13,120 14,611 16,473 19,939 24,337 29,339 34,382 37,652 40,646 44,314 46,928 52,62026 11,160 12,198 13,844 15,379 17,292 20,843 25,336 30,435 35,563 38,885 41,923 45,642 48,290 54,05227 11,808 12,879 14,573 16,151 18,114 21,749 26,336 31,528 36,741 40,113 43,195 46,963 49,645 55,47628 12,461 13,565 15,308 16,928 18,939 22,657 27,336 32,620 37,916 41,337 44,461 48,278 50,993 56,89229 13,121 14,256 16,047 17,708 19,768 23,567 28,336 33,711 39,087 42,557 45,722 49,588 52,336 58,30130 13,787 14,953 16,791 18,493 20,599 24,478 29,336 34,800 40,256 43,773 46,979 50,892 53,672 59,70340 20,707 22,164 24,433 26,509 29,051 33,660 39,335 45,616 51,805 55,758 59,342 63,691 66,766 73,40250 27,991 29,707 32,357 34,764 37,689 42,942 49,335 56,334 63,167 67,505 71,420 76,154 79,490 86,66160 35,534 37,485 40,482 43,188 46,459 52,294 59,335 66,981 74,397 79,082 83,298 88,379 91,952 99,607

Exemplo: amostra de tamanho n = 100

Nº defeitos (xi)

0 1 2 3 4 5 6 7 8 ...

Nº aparelhos (fi)

25 35 18 13 4 2 2 1 0 ...

Teste de Aderência pelo Método 2

Testar: H0 :Distribuição do nº de defeitos é Poisson

H1 : Tal não ocorre

...) 2, 1, 0, (r !

*)(

rerXPp

r

r

55,1100

1*7...25*0*

nfxx ii

Parâmetro da Poisson estimado por:

21201

10

55105515510

0 ,*!*),()(

,,

eeXPp 329,0*55,1

!1*)55,1()1( 55,1

55,11

1

eeXPp

X O p E (Oi – Ei)2 / Ei

0 25 0,212 21,2 0,618 1 35 0,329 32,9 0,134 2 18 0,255 25,5 2,206 3 13 0,132 13,2 0,003 4 4 0,051 5,1 5 2 9 0,016 1,6 7,2 0,450 6 2 0,004 0,4 7 1 0,001 0,1 100 1,0 100 3,474 = 2

= k - 1 - m = 5 - 1 - 1 = 3

Ei5 (exigência)

2. Teste de Independência (tabelas de contingência)

Exemplo: (Ex 01, p. 135-40, COSTA NETO)

Opiniões de homens e mulheres sobre determinado Projeto de Lei:

Opinião Sexo Favor Contra Indiferente Total

Homens 33 12 15 60 Mulheres 7 20 13 40

Total 40 32 28 100

H0 : Existe independência entre opinião e sexo

H1 : Tal não ocorre, ou seja, o sexo influencia na opinião

Opinião (Ho Verdadeiro) Sexo Favor Contra Indiferente Total

Homens 24 60 Mulheres 16 40

Total 40 32 28 100

Oij Eij = n*pi*pj Oij - Eij (Oij – Eij)2 / Eij

33 24,0 9,0 3,375 12 19,2 -7,2 2,700 15 16,8 -1,8 0,193 7 16,0 -9,0 5,063 20 12,8 7,2 4,050 13 11,2 1,8 0,289

100 100 15,670 = 2

Opinião Sexo Favor Contra Indiferente Total

Homens 33 12 15 60 Mulheres 7 20 13 40

Total 40 32 28 100 Opinião (Ho Verdadeiro)

Sexo Favor Contra Indiferente Total Homens 24 19,2 16,8 60 Mulheres 16 12,8 11,2 40

Total 40 32 28 100

= (r - 1) * (s - 1) = (2 - 1) * (3 - 1) = 2

98,5 93,6 101,2 106,4 106,0 108,7 89,1 98,8 105,596,5 91,5 90,8 95,1 96,1 89,4 97,2 98,0 100,798,4 98,1 106,7 99,6 99,1 97,4 109,9 104,3 111,9108,3 91,9 102,7 104,5 109,6 99,6 97,4 103,4 98,192,8 95,8 92,4 98,2 99,8 100,1 98,4 111,0 91,387,1 107,2 93,6 96,7 103,8 102,3 104,4 103,0 93,1103,5 101,6 95,3 98,8 100,7 102,8 100,7 95,4 109,4100,4 104,1 104,3 96,8 95,4 105,6 94,0 92,2 103,6

Testar: H0 :Distribuição de X é N(100;5)

H1 : Tal não ocorre

observado esperadomedia= 99,80 100--105 29 34,13dp= 5,43

Testar: H0 :Distribuição de X é exponencial - 1/=23

H1 : Tal não ocorre

Pr[a<X<b]=e-a- e-b

OBS ESP(0-7) 70(7-15) 58(15-25) 44(25-35) 0,118909 25(35-50) 22(50-70) 18(70-200) 13

250